4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic toK1,3. The kth power of a graphG, denoted byGk , is the graph with vertex set V (G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are C2 6 , C 2 8 and Tr , r 2, where Tr is the graph with 4r vertices ui, vi , xi , yi , 1 i r , such that for each i with 1 i r , {ui, vi , xi , yi} is a clique of Tr and xiui+1, yivi+1 ∈ E(Tr ) (mod r). We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K4 ×K2, C2 6 and C2 8 . © 2005 Elsevier B.V. All rights reserved.